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Multi antenna receivers are often deployed in cognitive radio systems for accurate spectrum sensing. However, correlation among signals received by multiple antennas in these receivers is often ignored which yield unrealistic results. In this paper, the effect of this correlation is accurately quantified by deriving analytical expressions for the average probability of detection. Alternative simpler expressions are also derived. These are done for Selection Combining (SC) and Switch and Stay (SSC) diversity techniques in dual arbitrarily correlated Nakagami-m fading channels. Then it is repeated for triple exponentially and identically correlated Nakagami-m fading channels with SC diversity technique. Analysis results show that the inter-branch correlation impacts the detector performance significantly, especially in deep fading scenarios. Also, SC outperforms SSC as expected. However, the difference between them becomes very small in low fading and highly correlated scenarios which, indicates that the simpler SSC scheme can as well be deployed in such situations.
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 1
Impact of Interbranch Correlation on Multichannel
Spectrum Sensing with SC and SSC Diversity
Combining Schemes
Salam Al-Juboori, Student Member, IEEE, Xavier Fernando, SMIEEE, Yansha Deng and
A. Nallanathan Fellow, IEEE,
Department of Electrical and Computer Engineering, Ryerson University, Toronto, Canada
Department of Informatics, King’s College London, London, WC2R 2LS, UK
School of Electronic Engineering and Computer Science, Queen Mary University of London, U.K
email: saljuboo@ryerson.ca, fernando@ee.ryerson.ca, yansha.deng@kcl.ac.uk, a.nallanathan@qmul.ac.uk
Abstract—Multi antenna receivers are often deployed in
cognitive radio systems for accurate spectrum sensing. However,
correlation among signals received by multiple antennas in these
receivers is often ignored which yield unrealistic results. In
this paper, the effect of this correlation is accurately quantified
by deriving analytical expressions for the average probability
of detection. Alternative simpler expressions are also derived.
These are done for Selection Combining (SC) and Switch and
Stay (SSC) diversity techniques in dual arbitrarily correlated
Nakagami-mfading channels. Then it is repeated for triple
exponentially and identically correlated Nakagami-mfading
channels with SC diversity technique. Analysis results show that
the inter-branch correlation impacts the detector performance
significantly, especially in deep fading scenarios. Also, SC
outperforms SSC as expected. However, the difference between
them becomes very small in low fading and highly correlated
scenarios which, indicates that the simpler SSC scheme can as
well be deployed in such situations.
Index Terms—Cognitive radio networks, spectrum sensing,
inter-branch correlation, diversity combining, selection
combining, switch and stay combining.
I. INTRODUCTION
P
ROTECTING the primary users from detrimental
interference from the secondary user signals is crucial in
cognitive radio systems. Accurate spectrum sensing is essential
for this. Simple schemes such as Energy Detectors (ED) are
widely used for this purpose that detect weak signals in noisy
channels as long as the noise power is known [1]. Accurate
spectrum sensing suffers from few issues, multipath fading and
shadowing being the leading causes. Multi antenna receivers,
with appropriate diversity combining schemes, are designed
to overcome these issues. Ideally, wireless channels seen by
the multiple antennas shall be independent to obtain the best
results from these diversity receivers [2]. However, often this is
not the case especially, when antennas are increasingly placed
closer to each other as the mobile units get smaller and more
demanding. Therefore, ignoring inter-branch correlation yields
inaccurate, especially overly optimistic, results. The effects
of multipath fading and correlation among antenna branches
heavily depend on the type of the diversity combining technique
employed. It is well known that Maximal Ratio Combining
scheme (MRC) is the optimal scheme which is also the
most complex linear diversity scheme. Equal Gain Combining
(EGC) diversity technique is a close competitor. Both the
MRC and EGC techniques require all or some knowledge
of the Channel State Information (CSI) [3]–[5]. Furthermore,
in these schemes each diversity branch must be equipped
with a single receiver that increases the system complexity.
Recently simpler combining schemes such as Switch and Stay
Combining (SSC) and Selection Combining (SC) are getting
popular due to their simplicity. These are especially useful in
cognitive radio networks. With the SC scheme, the receiver
simply selects the antenna with the highest received signal
power and ignores other antennas. Hence, signal combiners,
phase shifters or variable gain controllers are not required [3],
[6]. The SSC diversity technique is the least complex system
where no real combiner is required. The SSC selects a particular
antenna branch until its SNR drops below a predetermined
threshold [3]. Both SC and SSC schemes are required to
measure only the amplitude on each branch (in order to select
the highest one). Hence, they can be employed for both coherent
and non-coherent modulation schemes [3]. Different diversity
combining techniques have been studied in the literature. In [7],
averaging the probability of detection over fading channels with
Rayleigh, Nakagami-
m
and Rician distributions are studied, and
closed-form expressions for detector parameters were derived
for Nakagami-
m
channels with integer values of
m
. In [8]-[9],
alternative analytic approaches to that in [1] and [7] were given.
Furthermore, in [8], independent and identically distributed
(
i.i.d
) dual and
L
number of Rayleigh fading branches were
considered with SSC and SC diversities. Corresponding average
probability of detection expressions (
PD
) were also derived
for both techniques. In [10]-[11], closed-form expressions for
PD
were derived for
i.i.d.
diversity branches in Nakagami-
m
fading channels employing SC technique. In [12], closed-form
expressions of
PD
for
i.i.d
dual Nakagami-
m
fading branches
with SSC were derived for real and integer
m
values. In
our previous work [6], we have done an investigation on
the probability of detection for SC diversity with correlated
Nakagami-
m
fading branches. A review of prior works reveals
that correlated fading branches with SSC diversity is not studied
in the literature. However, because of the simplicity, the SSC
is particularly valuable for mobile stations that have limited
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 2
resource and power. This paper aims to fulfill that requirement.
In this paper, we extend our previous investigations by
considering SC and SSC diversity combining techniques with
identically correlated branches in Rayleigh and Nakagami-
m
fading channels.
Our contribution falls into two folds. First
We consider SC and SSC schemes with dual arbitrarily
correlated branches in Rayleigh and Nakagami-
m
fading
channels.
Then, we extend study of SC diversity to triple
exponentially correlated branches.
Corresponding novel expressions for average probability
of detection are derived for each case.
Alternative and more general and simpler expressions are
also derived for each case.
For SSC diversity, we derive an expression which can be
solved numerically to calculate the optimal SNR threshold
value in order to optimize the detector performance.
All our derived expressions do converge rapidly.
Secondly, to gain better insight
We do a performance comparison between the two
combining diversity techniques
Analysis results show that the inter-branch correlation
affects the detector performance significantly, especially
in deep fading scenarios.
SC outperforms SSC as expected however; the difference
between them becomes very small in low fading scenario
with highly correlation among antennas. This indicates
that the simpler SSC scheme can be substituted for the
SC scheme in these situations.
The rest of the paper is organized as follows. Section II
describes the system model. In section III, we study the
performance of SC scheme. In section IV, we study the
performance of SSC scheme. Section V describes simulation
and analysis results. Section VI concludes the paper.
II. SY ST EM MO DE L
We follow a binary hypothesis testing on the received signal
to declare the presence or absence of the primary user. For
this, we employ ED that is widely used in cognitive spectrum
sensing. Note that no priori information about the detected
signal is needed for ED [13], [14].
Let x(t)be the received observations data
x(t) = h s (t) + n(t),(1)
where,
h
is the complex channel gain amplitude coefficient,
assumed to be constant during the sensing time,
s(t)
is the
signal to be detected and,
n(t)
is the AWGN noise. This noise
is a low-pass Gaussian process with zero mean and variance
N0W
where,
N0
and
W
denote Power Spectral Density (PSD)
of the Gaussian noise and the signal bandwidth, respectively.
Two hypotheses are defined for the decision statistics.
Namely
H0
and
H1
, for the absence and the presence of
the primary user signal respectively, as follows:
x(t) = (n(t)under H0
h s (t) + n(t)under H1.(2)
The decision statistics is squared and integrated over time
T
at the ED. The output is written as
y,2
N0ZT
0|x|2(t) dt. (3)
The Probability Density Function (PDF) of the decision
statistics yis given by [8] and [9]
pY(y) =
1
2uΓ(u)yu1ey
2,under H0
1
2y
2γu1
2e2γ+y
2Iu12γ y,under H1
(4)
where
γ
denotes the signal-to-noise-ratio,
Γ(.)
is the the
Gamma function and,
Iν(.)
is the
νth
order modified Bessel
function of the first kind. The parameter
u
depends on the
time-bandwidth product. In (4), it is clear that the decision
statistics has a central chi-square distribution with
2u
degrees
of freedom
χ2
2u
in the absence of the primary user signal,
i.e. the received samples are noise only. However, it has a
non-central chi-square distribution
χ2
2u(ψ)
with
2u
degrees of
freedom and non-centrality parameter
ψ= 2γ
in the presence
of the primary user signal [8] and [9].
Let us define
λ
as the decision threshold. Then the probability
of false alarm (
PF
) and the probability of detection (
PD
) of
the ED can be written as
PF=P r (y > λ |H0),(5)
PD=P r (y > λ |H1).(6)
where
P r (.)
denotes the Cumulative Distribution Function
(CDF). Consequently, the probability of false alarm and
probability of detection in AWGN channel are given as [8] -
[9]
PF=Γu, λ
2
Γ(u),(7)
PD=Qup2γ, λ,(8)
where
Γ(., .)
and
Qu(., .)
denote the upper incomplete Gamma
function and generalized Marcum Q-function, respectively.
These detection probabilities are conditioned upon the channel
realization. Also, they represent instantaneous probability of
detection. Therefore, we need to integrate this instantaneous
probability of detection over the SNR’s PDF of the
corresponding fading channel (
pγ Div (γ)
) to obtain the average
probability of detection PD,D iv1.
PD,D iv =Z
0
Qup2γ, λpγ Div (γ) dγ . (9)
The expression in (9) will serve as a general expression for
the corresponding diversity channel.
Note that the probability of detection expression in (8) is
restricted to only integer values of
u
since the PDF of the
decision statistics in (4) is derived only for even numbers,
i.e.
2u
, as stated in [8]. However, when the alternative
Marcum-Q function is employed,
u
could be half-odd integer
1
False alarm probability is not a function of SNR as no signal is transmitted,
therefore it will remain unchanged as in (7).
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 3
(
u∈ {0.5,1,1.5,2,2.5,3, ...}
, i.e. not restricted to integer
values) [15]. Furthermore, the fading parameter
m
in Nakagami
channels might also be not restricted to integer values
depending on the mathematical method employed to solve the
integral in (9). This highlights the advantage of the alternative
expressions which we derive later using alternative Marcum-Q
function.
III. SC DIVERSITY WITH COR RE LATE D NAKAGAMI-m
FADING CHANNELS
In this section, we will derive the average probability of
detection for dual and triple Nakagami-
m
correlated fading
branches with SC diversity.
Selection Combining is a low complexity diversity technique,
as it chooses the highest SNR’s branch using the relationship
r=max {rl, l = 1,2, ... L}.(10)
Therefore it processes one branch a time. Consequently, no
phase knowledge is required.
The PDF of a univariate
r
-Nakagami-
m
variable is given
by [16]
fr(r) = 2
Γ(m)m
mr2m1em
r2, r 0(11)
where,
Γ(.)
denotes the Gamma function,
Ω = E[r2]/m =¯
r2
m
is the mean value of the variable
r
, and
m
(
m1/2
) is the
inverse normalized variance of
r2
, which describes the fading
severity.
We define the instantaneous SNR per symbol per channel
γl
as
γl=r2
lEs
N0
;
l[1,2, ... L]
;
Es
is the energy per symbol
and,
N0
is the PSD of the Gaussian noise. The average SNR
per branch is
¯γl=¯
r2
l
Es
N0
where,
¯
r2
l=E[r2
l]
is the expectation
of the channel envelop.
A. SC with Dual arbitrarily Correlated Branches
Using [[17], Eq. (20)] and, by assuming identical diversity
branches and by changing variables with some mathematical
simplification, the PDF of the output SNR for a dual SC
combiner under correlated Nakagami-
m
fading channels can
be obtained as
pγ SC (γ) = 2
Γ(m)m
¯γm
γm1exp m γ
¯γ
×h1Qmp2a ρ γ, p2a γi, γ 0
(12)
where,
ρ
denote the correlation coefficient between the two
fading envelopes, and
a=m
¯γ(1ρ)
. Please see Appendix A for
detailed derivation.
By substituting (12) into (9), the average probability of
detection for dual correlated SC’s diversity branches (
PD,S C,2
)
is obtained as
PD,S C,2=2
Γ(m)m
¯γm
[IAIB],(13)
where
IA=Z
0
Qup2γ , λγm1exp
¯γdγ, (14)
and
IB=Z
0
Qup2γ , λQmp2aργ, p2
×γm1e
¯γdγ.
(15)
Note this lengthy expression consists of two integrals,
IA
and
IB. We solve them separately. Please see Appendix B.
Hence, the average probability of detection for dual
SC receiver under correlated identical Nakagami-
m
fading
branches (restricted to integer uand mvalues) is
PD,S C,2=2
Γ(m)m
¯γm"1
2m1(G1+η
2
u1
X
n=1
1
n!λ
2n
×1F1m;n+ 1; λ¯γ
2 (m+ ¯γ)
X
n=0
X
i=0
i+m1
X
k=0
Γu+n, λ
2
Γ(u+n)n!
×ai+kρi(i+k+m+n1)!
ci+k+m+ni!k!#,
(16)
where
c=1 + a(ρ+ 1) + m
¯γ
and
G1
for integer
m
values
is
G1=2m1(m1)!
m
¯γ2m¯γ
m+ ¯γeλ
2
m
mγ
×"m+ ¯γ
¯γm
m+ ¯γm1
Lm1λ¯γ
2 (m+ ¯γ)
+
m2
X
n=0 m
m+ ¯γn
Lnλ¯γ
2 (m+ ¯γ)#.(17)
Here
Ln(.)
denotes Laguerre polynomial of
n
-degree [18],
and
1F1(., .;.)
denotes the Confluent Hypergeometric function.
This is defined in [[19], Eq. (15.1.1)] as
1F1(a1, b1;x) = Γ(b1)
Γ(a1)
X
i=0
Γ(a1+i)xi
Γ(b1+i)i!.(18)
Note (16) reduces to dual correlated Rayleigh fading branches
for
m= 1
. It’s worthwhile to mention that for
i.i.d.
diversity
branches, (16) reduces to [[9], Eq.(7), [8], Eq. (20)] multiplied
by 2 (not exceeding unity)). The latter expression was derived
for the average probability of detection in flat fading. Hence
we have improved the detection performance and derived (16)
to serve as a proof.
B. Alternative Expression for PD,SC,2
Despite the fact that
Qu2γ, λ
portion of the second
integral
IB
in (13) is evaluated for
u
values not-restricted to
integer, (16) is still restricted to integer values. This is because,
the first integral
IA
in (13) is only valid for integer
u
and
m
values. In this section, we derive a more general and simpler
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 4
alternative expression for (16) that is not restricted to integer
uvalues. Please see Appendix C for the derivation.
PD,S C,2= 1 2m
¯γm
eλ
2
×"1
dm
X
n=uλ
2n1
Γ(n+ 1)1F1m; 1 + n;λ
2d
1
Γ(m)
X
n=u
X
i=0
i+m1
X
k=0 λ
2nΓ(m+i+k)ai+kρi
Γ(n+ 1)cm+i+ki!k!
×1F1m+i+k; 1 + n;λ
2c#.
(19)
For
m= 1
, (19) reduces to the average probability of
detection with dual correlated Rayleigh fading branches and,
with
ρ= 0
to
i.i.d
dual Rayleigh fading branches given in (
[8], Eq. (30)).
Fortunately, the error resulting from truncating the
infinite series in (19) is upper bounded by the Confluent
Hypergeometric function defined in (18). Since this function
is monotonically decreasing with
i, k
and
n
for given values
of
m
,
λ
and
¯γ
[20], the number of terms (
Nn
and
Ni
) that
required five digit accuracy could be calculated. These numbers
are shown in Table I for different values of ρand m.
It’s worthwhile to mention that several solutions for integrals
involving the Marcum Q-function are available in literature
[21]–[25]. However, our case of study in (15) solves a different
and more complicated integral which involves a product of
two Marcum Q-functions. These solutions are introduced in
(57), (69) in Appendices B and C, respectively. To the best of
knowledge, we believe that this solution is new in literature.
Finally, we’d like to mention that the solutions introduced
in expressions (16) and (19) present a clear advantage
over the numerical integration approach showed in (13)
since a numerical integration is rather long and often gives
approximated result. Furthermore, although expressions in
(16) and (19) involve nested infinite series, they are either
upper bounded by a monotonically decreasing confluent
hypergeometric function or by an upper incomplete gamma
function. Note that the latter could also be represented by a
monotonically decreasing confluent hypergeometric function
using [[28], Eq. (1.6)]. Consequently, these infinite series terms
converge rapidly as we discussed earlier in Table I.
C. SC with Triple Correlated Branches
In this section, we consider triple correlated diversity
branches. We start from PDF of the fading envelope for
trivariate Nakagami-
m
channels given in [[26], Eq. (8)]. Then,
by changing variable and by assuming identical branches
(
¯γ= ¯γ1= ¯γ2= ¯γ3
, and the same fading parameter
m
), the
PDF of the output SNR for triple SC exponentially correlated
Nakagami-mbranches can be derived. This is shown below
pγ SC,3(γ) = Σ1m
Γ(m)
X
i=0
X
j=0
|p1,2|2i|p2,3|2j
pi+m
1,1pi+j+m
2,2pj+m
3,3
×1+ Θ2+ Θ3]
Γ(m+i) Γ(m+j)i!j!,
(20)
where
Σ1
is the inverse of the correlation matrix,
pi1,j1(i1, j1= 1,2,3)
being its entries and
Θ1,Θ2
and
Θ3
are
Θ1=p1,1m
¯γi+m
γi+m1ep1,1m
¯γγ
×γi+j+m, p2,2m
¯γγγj+m, p3,3m
¯γγ,
(21)
Θ2=p2,2m
¯γi+j+m
γi+j+m1ep2,2m
¯γγ
×γi+m, p1,1m
¯γγγj+m, p3,3m
¯γγ,
(22)
Θ3=p3,3m
¯γj+m
γj+m1ep3,3m
¯γγ
×γi+m, p1,1m
¯γγγi+j+m, p2,2m
¯γγ,
(23)
respectively. Here
γ(a, x)
denotes the lower incomplete
gamma function with
γ(a, x) = Rx
0etta1dt
([18],
Eq.(8.350/1)).
In exponentially correlated model, the diversity antennas are
equispaced. Therefore, the correlation matrix can be written as
Σi1,j1ρ|i1j1|
[27]. Hence, the inverse correlation matrix
Σ1is tridiagonal and can be written as
Σ1=1
ρ21
1ρ0
ρ(ρ2+ 1) ρ
0ρ1
,(24)
where ρdenotes the correlation coefficient.
We have made an assumption of identical average SNRs in
all three branches above. This assumption is reasonable if the
diversity channels are closely spaced and, their gains as well
as noise powers are equal [3].
The average probability of detection for triple SC diversity
Nakagami-
m
correlated branches with integer
u
is derived as
below. See Appendix D for details.
PD,S C,3=Σ1m
Γ(m)eλ
2
X
n=0
n+u1
X
k=0
X
i=0
X
j=0
"λ
2k¯γ
mn|p1,2|2i|p2,3|2j
pi+m
1,1pi+j+m
2,2pj+m
3,3Γ(m+i)Γ(m+j)
×pi+m
1,1pi+j+m
2,2pj+m
3,3Γ(2i+ 2j+ 3m+n)
p11 +p2,2+p3,3+¯γ
m(2i+2j+3m+n)i!j!k!n!
×1+ Ξ2+ Ξ3)#.(28)
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 5
Ξ1=
F22i+ 2j+ 3m+n; 1,1; i+j+m+ 1, j +m+ 1; p2,2
p11 +p2,2+p3,3+¯γ
m
,p3,3
p11 +p2,2+p3,3+¯γ
m
(i+j+m) (j+m),(25)
Ξ2=
F22i+ 2j+ 3m+n; 1,1; i+m+ 1, j +m+ 1; p1,1
p11 +p2,2+p3,3+¯γ
m
,p3,3
p11 +p2,2+p3,3+¯γ
m
(i+m) (j+m),(26)
Ξ3=
F22i+ 2j+ 3m+n; 1,1; i+m+ 1, i +j+m+ 1; p1,1
p11 +p2,2+p3,3+¯γ
m
,p2,2
p11 +p2,2+p3,3+¯γ
m
(i+m) (i+j+m).(27)
Here,
Ξ1,Ξ2
and
Ξ3
are as given in (25), (26) and (27) at the
top of next page, respectively, and
F2α3;β3, β0
3;γ3, γ0
3;x, y
denotes the Hypergeometric function of two variables defined
in [[18], Eq. (9.180.2)]. Note, for
m= 1
, (28) reduces to triple
correlated Rayleigh fading branches.
D. General Expression for Triple Branches
In this section, we will derive a general and simpler
alternative expression to (28), where both
u
and
m
are not
restricted to integer values. See Appendix E for details.
The average probability of detection for triple SC
Nakagami-
m
correlated branches for not restricted
u
or
m
integer values is:
PD,S C,3=Σ1m
Γ(m)
X
n=0
X
i=0
X
j=0 ¯γ
mnΓu+n, λ
2
Γ(u+n)
×|p1,2|2i|p2,3|2jΓ(2i+ 2j+ 3m+n)
Γ(m+i) Γ(m+j)i!j!n!
×1+ Ξ2+ Ξ3)
p11 +p2,2+p3,3+¯γ
m2i+2j+3m+n.(29)
where
Ξ1,Ξ2
and
Ξ3
are given in (25), (26) and (27),
respectively. As before, for
m= 1
, (29) reduces to triple
correlated Rayleigh fading branches. It’s worthwhile to
mention that the Hypergeometric function of two variables
F2α3;β3, β0
3;γ3, γ0
3;x, y
appears in (28) and (29) converges
only for
|x|+|y|<1
[18], where
|.|
denotes absolute.
Fortunately, this is the case in our above derived equations.
IV. DUAL CORRELATE D NAKAGAMI-mCHANNELS WITH
SSC DIVERSITY
The SSC receiver selects a particular diversity branch
until its SNR drops below a predetermined threshold value.
Hence SSC’s technique is similar to its counterpart SC but.
Nevertheless, the SSC receive does not need to continuously
monitor the SNR of each branch. Therefore, the SSC
is considered as the least complex
2
diversity combining
technique [3].
2
Other diversity combining techniques such EGC and MRC process more
than one branch and require the channel state knowledge of some or all the
branches [3].
Starting from [[3], p.437, Eq. (9.334)], the SNR’s PDF for
a dual and identical correlated Nakagami-
m
fading channels
with SSC combiner is
pγ SS C (γ) = (A(γ)γγT
A(γ) + m
¯γmγm1
Γ(m)exp m γ
¯γγ > γT,
(30)
where
γT
denotes a predetermined switching threshold and
A(γ)is given in [[3], p.437, Eq. (9.335)] as
A(γ) = m
¯γmγm1
Γ(m)exp m γ
¯γ
×h1Qmp2a ρ γ, p2a γTi,
(31)
where
a=m
¯γ(1ρ)
and
Qm(., .)
denotes generalized Marcum
Q-function.
The average probability of detection for dual correlated
Nakagami-mfading branches with SSC diversity (PD,SS C,2)
is obtained by substituting (30) into (9) and then using the
definition R
afdx=R
0fdxRa
0fdx, which yields
PD,S SC,2=1
Γ(m)m
¯γm
[IAIBIC](32)
with
IA= 2 Z
0
Qup2γ, λγm1exp m γ
¯γdγ, (33)
IB=Z
0
Qup2γ, λQmp2a ρ γ , p2a γT
×γm1exp m γ
¯γdγ, (34)
and
IC=ZγT
0
Qup2γ, λγm1exp m γ
¯γdγ. (35)
Before deriving an expression for the probability of detection
PD,S SC,2
, it is worthy to investigate (32) for the following
two special cases of threshold values.
Case I: γT= 0
If
γT= 0
, we have
Qm2a ρ γ, 2a γT= 1
and the third
term
IC
vanishes, consequently (32) reduces to single branch
detection as
PD,S SC,2=1
Γ(m)m
¯γmZ
0
Qup2γ, λ
×γm1exp m γ
¯γdγ.
(36)
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 6
Case II: γT→ ∞
If
γT→ ∞
, we have
Qm2a ρ γ, 2a γT= 0
,
consequently
IB
vanishes and only
IC
is subtracted from
IA
.
This results in single branch detection as in (36). Therefore, care
must be taken to choose a sensible threshold value. Otherwise,
the diversity technique might become useless.
The average probability of detection for dual correlated SSC
receiver with Nakagami-
m
fading branches where
u
and
m
are restricted to integer values is given in (37) at the top of
next page. Please see Appendix F for detailed derivation.
Note that for
m= 1
, (37) reduces to dual Rayleigh correlated
fading branches, and for
ρ= 0
it reduces to dual
i.i.d.
Nakagami-mfading branches detection.
A. Alternative Solution
The expression
PD,S SC,2
in (37) involves many infinite
series representations. Some of their upper bounds (number of
terms) are dependent on the preceded one. As an example the
upper bound of the second sum (
Pj+u1
k=0 (.)
) depends on the
number of terms
(N)
needed for convergence of the previous
series. Fortunately, it will not be very difficult to find the
number of terms for convergence (with five digit accuracy).
However, time for numerical implementation will be rather
long. Therefore, we will derive an alternative more general
and simpler expression
PD,S SC,2
with less number of infinite
series representations.
The average probability of detection where
u
is not restricted
while (
m1
) is restricted to integer values is given in (38)
at the top of next page. See Appendix G for the derivation.
Note, for
m= 1
, (38) reduces to that of a dual SSC receiver
with Rayleigh correlated fading branches. For,
ρ= 0
it reduces
to the PDF of the dual
i.i.d.
Nakagami-
m
fading branches
detection.
Interestingly, the three terms in (38) contain the upper
incomplete gamma function in addition to the lower incomplete
gamma function in the last term. In fact, we can represent
both these functions by the monotonically decreasing confluent
hypergeometric function using [[19], Eq. (6.5.12)] and [[28],
Eq. (1.6)] for lower and upper incomplete gamma functions,
respectively. Consequently the infinite series terms in (38)
converges rapidly.
B. Optimal Threshold (γ
T)
Optimal threshold
γ
T
is defined as the value of the SNR
that maximizes the probability of detection. We maximize the
probability of detection by selecting an appropriate SNR for
SSC switching. Probability of false alarm is fixed since it’s
a function of the decision threshold
λ
and not a function of
SNR, as shown in (7). Constant False Alarm Rate (CFAR) is
a well-known technique that is often employed in cognitive
spectrum sensing. In this technique and using (7), a decision
threshold is calculated for fixed probability of false alarm. Then
the corresponding probability of detection is calculated using
(8) for optimal SNR. We have derived an expression for this
optimal threshold given in (39) at the top of this page. This is
done by differentiating
PD,S SC,2
in (32) with respect to
γT
and solving
∂γ
TPD,S SC,2= 0
for
γ
T
. See Appendix H for
Table I: Terms required for five digits accuracy
PD,S C,2:
˜
EN
, u = 2, P F= 0.01,¯γ= 20 dB
ρm= 1
Nn,Ni
m= 2
Nn,Ni
m= 3
Nn,Ni
m= 4
Nn,Ni
0 15,1 15,1 15,1 15,1
0.2 15,3 15,3 15,2 15,1
0.4 15,3 15,2 15,1 15,1
0.6 15,3 15,5 15,4 15,4
0.8 15,4 15,5 15,6 15,7
details. Using Matlab, we can obtain the optimal threshold by
evaluating (39) numerically for
∂γ
TPD,S SC,2= 0.
V. SIMULATION AND ANALYS IS RE SU LTS
The energy detector employed in spectrum sensing is
mainly characterized by the probability of false alarm
PF
and
probability of detection
PD
. In this section we study the impact
of the correlation among antenna diversity branches on
PD
(equivalently probability of miss detection
PDm = 1 PD
) as
a performance metric using the derived expressions in previous
sections. To this end, we produce Complementary Receiver
Operating Characteristic (CROC) graphs (
PDm
versus
PF
)
for SC and SSC diversity techniques in Nakagami-
m
fading
channel.
First, we plot the probability of miss detection with the
corresponding threshold
λ
for
u= 2
,
¯γ= 20
dB,
m(1,4)
and,
ρ(00.8)
for different values of
PF
using (7). Through
Monte Carlo simulation, we obtain the CROC curves for SC
and SSC. We then compare the simulation results with the
analytical curves obtained from derived expressions.
In Figure. 1, we plot the CROC graphs for
L= 2, m =
1,¯γ= 20
dB and
ρ= 0.8
. Results are obtained for SC and
SSC using both the derived expressions (analytical) and by
Monte Carlo simulation. For SC diversity, both these curves
are almost in a perfect match. However, reader may observe a
very small difference between analytical and simulation curves
for SSC diversity. This is due to the inaccuracy arising from
rounding off the infinite series and calculating the optimal
threshold .
In Figure 2, we plot the CROC graphs for SC with
¯γ=
20
dB,
m(1,4)
and,
ρ(0 0.8)
. For each value of
fading severity
m
, one can clearly notice the degradation in the
probability of detection due to the correlation among diversity
branches. For instance, let us consider the case
m= 1
and
constant
PF= 0.01
as in Figure 2a. The corresponding
PDm
for
ρ= 0.8
is almost four times its value for
ρ= 0
(no
correlation). Similar result could be observed in Figure 2b,
however, the increment ratio is now much more larger. However,
as
m
increases (low fading environment), correlation effect is
compensated for, resulting in higher probability of detection
(equivalently, low probability of miss detection). Thus, the rate
of correlation compensation due to good channel is higher than
the correlation impact on probability of detection.
For easy and better comparison between SC and SSC and
their performance in combating the correlation, we plot CROC
graphs in Figure 3 for
¯γ= 20
dB,
m(1,4)
and,
ρ(0,0.8)
.
As before, one can notice the impact of the correlation between
fading branches on the probability of detection. This impact is
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 7
PD,S SC,2=1
Γ(m)m
¯γm
eλ
2"2
X
j=0
j+u1
X
k=0 λ
2k(j+m1)!
j!k!¯γ
¯γ+mj+m
X
n=0
n+u1
X
q=0
X
i=0
i+m1
X
k=0 λ
2q
×ai+kρi(m+n+i1)!
a ρ +m
¯γ+ 1m+n+ii!k!n!q!
ea γTγk
T
X
n=0
n+u1
X
q=0 λ
2q1
n!q!¯γ
¯γ+mm+n
γm+n, γT¯γ+m
¯γ#(37)
PD,S SC,2=1
Γ(m)m
¯γm"4
X
j=0
Γu+j, λ
2Γ(m+j)
Γ(u+j)1 + m
¯γm+jj!
X
n=0
X
i=0
i+m1
X
k=0
Γu+n, λ
2Γ(m+n+i)γk
TeTai+kρi
Γ(u+n)m
γ++ 1m+n+in!i!k!
X
p=0
Γu+p, λ
2
Γ(u+p)p!¯γ+m
¯γ(m+p)
γm+p, γT¯γ+m
¯γ#.
(38)
∂γ
T
PD,S SC,2=1
Γ(m)m
¯γm"p2a γ
Te
T
X
k=0
am+2k1ρk
Γ(m+k)2m+kk!γ
T
k(G0
1+1
2
u1
X
n=1 λ
2nΓ (m+k)
a+1
2m+kn!
×1F1m+k;n+ 1; λ
2 (a+ 1))Qup2γ
T,λγ
T
m1exp m γ
T
¯γ#.
(39)
10-4 10-3 10-2 10-1 100
Probabilty of False Alarm PF
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Probability of Miss Detection PDm
SC Analytic
SC Simulation
SSC Analytic
SSC Simulation
Figure 1: Analytic (solid) versus simulation (dashed) results for
SC and SSC derived expressions with
L= 2, m = 1
,
¯γ= 20
dB and ρ= 0.8.
compensated by good channel. Furthermore, results in Figure 3
show that SC outperforms SSC. This is a well proven fact in the
literature. In fact, performance difference is more pronounced
for uncorrelated (
ρ= 0
) and high
m
values. However, we may
notice that as the correlation increases between the branches,
the performance of both SC and SSC schemes becomes more
comparable. This is especially true for high mvalues.
Figure 4 shows probability of miss detection versus
correlation for
¯γ= 20
dB,
m(1,4)
and,
PF= 0.01
for both
SC and SSC diversity techniques. Another interesting behaviour
that could be observed from this figure. As
m
increases
(equivalently, fading decreases), less significant deterioration
in probability of detection is observed due to correlation. In
other words, the loss in diversity gain due to correlation gets
lower as mincreases.
To gain better insight about this behaviour, let us discuss it
with more details. Figure 4a shows clearly this interesting
behaviour. The curve for
m= 1
in Figure 4a has an
average high positive slope. Consequently, the probability
of detection degrades rapidly as correlation increases. As
m
increases, corresponding curves get flattened (slope decreases).
Consequently, probability of detection degrades slowly as
correlation increases. This can be attributed to the fact that
already the
PD
values are high due to low fading. On the other
hand, for small
m
-values (deep fading), correlation significantly
deteriorates the probability of detection which is already poor.
A similar behaviour could be observed in the SSC shown in
Figure 4b. Therefore, we conclude the following. In a deep
fading scenario, the inter-branch correlation is a crucial factor
and its effects must be incorporated in any spectrum sensing
model. By contrast, in a low fading environment (those having
large values of
m
), the effect of such correlation may be ignored
without much impact.
VI. CONCLUSION
In this work, we have investigated the impact of correlation
among diversity fading branches in multi-antenna cognitive
radio spectrum sensing networks. A unified performance
analysis was presented for the probability of detection of
SC and SSC diversity combining receivers with arbitrary
and exponential correlation among fading branches. Exact
expressions were derived for the probability of detection for
each case. Our result show that the correlation among diversity
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 8
(a) m= 1 (Rayleigh)
10-4 10-3 10-2 10-1 100
Probability of False Alarm "PF"
10-14
10-12
10-10
10-8
10-6
10-4
10-2
Probability of Miss Detection " PDm"
= 0
= 0.2
= 0.4
= 0.6
= 0.8
(b) m= 4
Figure 2: SC dual correlated Nakagami-mbranches with ¯γ= 20 dB for different ρvalues.
10-4 10-3 10-2 10-1 100
Probability of False Alarm "PF"
10-14
10-12
10-10
10-8
10-6
10-4
10-2
Probability of Miss Detection " PDm"
SC: = 0
SC: = 0.8
SSC: = 0
SSC: = 0.8
(a) m= 1 (Rayleigh)
10-4 10-3 10-2 10-1 100
Probability of False Alarm "PF"
10-14
10-12
10-10
10-8
10-6
10-4
10-2
Probability of Miss Detection " PDm"
SC: = 0
SC: = 0.8
SSC: = 0
SSC: = 0.8
(b) m= 4
Figure 3: SC/SSC dual correlated Nakagami-mbranches comparison with ¯γ= 20 dB for ρ= 0 (solid) and 0.8 (dashed).
fading branches causes an adverse impact on the probability
of detection, which cannot be ignored especially under severe
fading conditions. Consequently, an increase in the interference
rate between the primary user and secondary user is observed
by three times its rate when independent fading branches
is assumed. Our investigations reveal that for low fading
environment (large
m
-values), correlation effect may be ignored.
Furthermore, at low fading and highly correlated environments,
SSC which is simpler scheme performs as good as SC which
is a more complex scheme.
APPENDIX A
DERIVATION OF (12)
Using ([17], (20)), the PDF of SC’s output of dual identical
correlated Nakagami fading branches is
pγ SC (r) = 4mmr2m1
Γ(m)Ωmexp m r2
×h1Qmp2ρA r, 2A ri,
(40)
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Correlation " "
0
2x10-3
4x10-3
6x10-3
8x10-3
10x10-3
12x10-3
14x10-3
16x10-3
18x10-3
Probability of Miss Detection " PDm"
m = 1
m = 2
m = 3
m = 4
(a) SC
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Correlation " "
0
5x10-3
10x10-3
15x10-3
20x10-3
25x10-3
Probability of Miss Detection " PDm"
m = 1
m = 2
m = 3
m = 4
(b) SSC
Figure 4: Probability of miss detection versus correlation with
¯γ= 20
dB,
PF= 0.01
and different fading severity for SC and
SSC.
where A=qm
Ω(1ρ).
Changing variables using pγ(γ) = prqγ
¯γ
2¯γ γ
[3] yields
pγ SC (γ) =
4mmqγ
¯γ2m1
2q¯γ γ
Γ(m)Ωm
exp
mqγ
¯γ2
×"1Qm p2ρAsγ
¯γ,2Asγ
¯γ!#,
(41)
Simplifying, (41) becomes
pγ SC (γ) = s
¯γ γ !2mmqγ
¯γ2m1
Γ(m)Ωmexp m γ
¯γ
×"1Qm p2ρAsγ
¯γ,2Asγ
¯γ!#,
(42)
Substituting A=qm
Ω(1ρ)and simplifying, yields
pγ SC (γ) = 1/2+m1/2
¯γ1/2γ1/2
2mmγm1/2
Γ(mγm1/2mexp m γ
¯γ
"1Qm p2ρrm
Ω(1 ρ)sγ
¯γ,2rm
Ω(1 ρ)sγ
¯γ!#,
(43)
Simplifying
pγ SC (γ) = 2mm
Γ(mγmγ1mexp m γ
¯γ
×"1Qm s2
¯γ(1 ρ)γ, s2m
¯γ(1 ρ)γ!# (44)
Simplifying and rearranging, this concludes the derivation.
APPENDIX B
EXPRESSION FOR DUAL S C
In this appendix, we derive the expression in (16).
1) Evaluating
IA
in (14): Introducing changing variable
x=2γ, we can derive
IA=1
2m1Z
0
Qux, λx2m1exp m x2
2 ¯γdx
| {z }
I
.
(45)
Using [[29], Eq. (29)], we write
Z
0
Qu(α x, β )xqep2x2
2dxGu
=Gu1+
Γq+1
2β2
2u1eβ2
2
2 (u1)! p2+α2
2q+1
2
×1F1q+ 1
2;u;β2
2
α2
p2+α2, q > 1,(46)
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 10
we can solve
I
by evaluating
Gu
recursively for
q > 1
and
restricted uinteger values as
Gu=Gu1+Au1Fu
=Gu2+Au2Fu2+Au1Fu1
.
.
.
=G1+
u1
X
n=1
AnFn+1.
(47)
where Anand Fnare given as
An=1
2 (n!) p2+α2
2q+1
2
Γq+ 1
2β2
2n
eβ2
2,(48)
Fn=1F1q+ 1
2;n;β2
2
α2
p2+α2,(49)
and
1F1(., .;.)
as defined previously in (18). Hence, we solve
(45) to obtain
IA=1
2m1G1+η
2
u1
X
n=1
1
n!λ
2n
×1F1m;n+ 1; λ¯γ
2 (m+ ¯γ),
(50)
where
η= Γ(m)2 ¯γ
mγmeλ
2
and
G1
can be obtained by
evaluating the following integral containing the first order of
Marcum Q-function Q(., .)for integer mvalues as
G1=Z
0
Qx, λx2m1em x2
2 ¯γdx. (51)
Using [[29], Eq. (25)], we evaluate
G1
for integer
m
values
as in (17).
2) Evaluating
IB
in (15): Using the alternative canonical
Marcum Q-function representations for
Qu2γ, λ
given
in [15] for not restricted to integer values of uas
Qup2γ, λ=
X
n=0
γneγΓu+n, λ
2
Γ(u+n)n!,(52)
and the alternative representation given in [[3], Eq. (4.74)] for
restricted minteger values as
Qm(α1, β1) =
X
i=0
exp α2
1
2α2
1
2i
i!
×
i+m1
X
k=0
exp β2
1
2β2
1
2k
k!,
(53)
therefore, Qm2a ρ γ, 2a γcould be written as
Qmp2aργ, p2=
X
i=0
i+m1
X
k=0
ai+kρi
i!k!e(ρ+1) γi+k.
(54)
Then by substituting (52) and (54) into (15) with some
simplification we derive
IB=
X
n=0
X
i=0
i+m1
X
k=0
Γu+n, λ
2
Γ(u+n)n!
ai+kρi
i!k!
×Z
0
γi+k+m+n1eγc dγ
| {z }
I
,
(55)
where
c= 1 + a(ρ+ 1) + m
¯γ
. Now the next task is solving
the integral
I
in (55). For this we use [[18], Eq. (3.351/3)] and
satisfying the condition therein,
Z
0
xp1eµ1xdx=p1!µ1p11[Re µ1>0] .(56)
Hence (55) becomes
IB=
X
n=0
X
i=0
i+m1
X
k=0
Γu+n, λ
2
Γ(u+n)
(i+k+m+n1)!ai+kρi
ci+k+m+nn!i!k!
(57)
Substituting (50) and (57) into (13), this concludes the
derivation.
APPENDIX C
ALTER NATI VE EXPRESSION FOR DUA L SC
In this Appendix, we derive (19). Using the alternative
expression for Marcum Q-function given in [[3], Eq.(4.63)],
where
u
is not restricted to integer values, we can write
Qu2γ, λas
Qup2γ , λ= 1 e2γ+λ
2
X
n=u λ
2γ!n
Inp2λγ.
(58)
Then substituting (12) in (9) and using the definition of the
PDF as
Z
0
pγ(γ) dγ= 1,(59)
with simplification, we can derive
PD,S C,2= 1 [IAIB],(60)
where
IA=2
Γ(m)m
¯γmZ
0
γm1e2γ+λ
2
X
n=u λ
2γ!n
×Inp2λ γ exp m γ
¯γdγ,
(61)
and
IB=2
Γ(m)m
¯γmZ
0
γm1e2γ+λ
2
X
n=u λ
2γ!n
×Inp2λγexp
¯γQmp2aργ, p2dγ, γ 0.
(62)
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 11
1) Evaluating
IA
in (61): Simplifying and rearranging (61),
we derive
IA=2
Γ(m)m
¯γm
eλ
2
X
n=uλ
2n
2
×Z
0
γmn
21eγ(1+ m
¯γ)Inp2λ γ dγ.
(63)
Using [[18], Eq. (6.643/2)] given as
Z
0
xµ1
2eα x I2ν2βxdx=
Γµ+ν+1
2
Γ(2 ν+ 1) β1eβ2
2ααµMµ,ν β2
α,
Re µ+ν+1
2>0,(64)
where Mµ,ν (.)denotes Whittaker function given by [18]
Mµ,ν (z) = zν+1
2ez
21F1νµ+1
2; 1 + 2ν;z,(65)
with some simplification and rearranging, the solution of (63)
can be derived as
IA=2
dmm
¯γm
eλ
2
X
n=uλ
2n1
Γ(n+ 1)
×1F1m; 1 + n;λ
2d,
(66)
where d=¯γ+m
¯γ.
2) Evaluating
IB
in (62): Simplifying and rearranging (62),
we derive
IB=2
Γ(m)m
¯γm
eλ
2
X
n=uλ
2n
2Z
0
γmn
21ed γ
×Inp2λ γ Qmp2a ρ γ, p2a γ dγ.
(67)
Using (54) with simplification and rearranging, we write (67)
as
IB=2
Γ(m)m
¯γm
eλ
2
X
n=u
X
i=0
i+m1
X
k=0 λ
2n
2ai+kρi
i!k!
×Z
0
γmn
2+i+k1eγc dγ,
(68)
where
c= 1 + a(ρ+ 1) + m
¯γ
. Similarly, implementing same
procedures as (66), the solution of (68) can be given as
IB=2
Γ(m)m
¯γm
eλ
2
X
n=u
X
i=0
i+m1
X
k=0 λ
2nai+kρi
i!k!
×Γ(m+i+k)
Γ(n+ 1) cm+i+k1F1m+i+k; 1 + n;λ
2c.
(69)
Substituting (66) and (69) into (60), this concludes the
derivation.
APPENDIX D
EXPRESSION FOR TRIPLE SC
In this section, we drive (28). Using (53) and substituting
(20) into (9), we drive the average probability of detection as
PD,S C,3=Σ1m
Γ(m)eλ
2
X
n=0
n+u1
X
k=0
X
i=0
X
j=0
×|p1,2|2i|p2,3|2jλ
2k
pi+m
1,1pi+j+m
2,2pj+m
3,3Γ(m+i) Γ(m+j)i!j!k!n!
×Z
0
γneγ1+ Θ2+ Θ3] dγ
| {z }
IA
.
(70)
Substituting (21), (22) and (23) into (70), the integral part
IA
in (70) becomes
IA=p1,1m
¯γi+m
Ia1+p2,2m
¯γi+j+m
Ia2
+p3,3m
¯γj+m
Ia3,
(71)
where
Ia1=Z
0
γi+m+n1eγ(p11 m
¯γ+1)(72)
×γi+j+m, p2,2m
¯γγγj+m, p3,3m
¯γγdγ.
Ia2=Z
0
γi+j+m+n1eγ(p2,2m
¯γ+1)(73)
×γi+m, p1,1m
¯γγγj+m, p3,3m
¯γγdγ.
Ia3=Z
0
γj+m+n1eγ(p3,3m
¯γ+1)(74)
×γi+m, p1,1m
¯γγγi+j+m, p2,2m
¯γγdγ.
Each integral in (71) could be written as
I=Z
0
xaebxγ(d1, c1x)γ(d2, c2x) dx. (75)
Using [[30], Eq. (10)], we write (71) as
IA=¯γ
mnpi+m
1,1pi+j+m
2,2pj+m
3,3Γ(2i+ 2j+ 3m+n)
p11 +p2,2+p3,3+¯γ
m(2i+2j+3m+n)
×1+ Ξ2+ Ξ3),
(76)
where
Ξ1
,
Ξ2
and
Ξ3
are in (25), (26) and (27), respectively.
Substituting (76) into (70), this concludes the derivations.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 12
APPENDIX E
GEN ER AL EXPRESSION FOR TRIPLE SC
In this section, we derive the expression in (29). Using (52)
and substituting (20) into (9) we derive
PD,S C,3=Σ1m
Γ(m)
X
n=0
X
i=0
X
j=0
Γu+n, λ
2
Γ(u+n)
×|p1,2|2i|p2,3|2j
pi+m
1,1pi+j+m
2,2pj+m
3,3Γ(m+i) Γ(m+j)i!j!n!
×Z
0
γneγ1+ Θ2+ Θ3] dγ
| {z }
IA
.(77)
Following same procedures in (71)-(76), then substituting (76)
into (77), this concludes the derivation.
APPENDIX F
EXPRESSION FOR DUAL SSC
In this section, we will derive the expression in (37) by
evaluating PD,SS C,2in (32) as follows.
1) Integral
IA
in (33): Using Marcum Q-function alternative
representation (53), we rewrite (33) as
IA= 2 eλ
2
X
j=0
j+u1
X
k=0 λ
2k1
j!k!
×Z
0
γj+m1exp γ(1 + m
¯γ)dγ.
(78)
Using (56) and satisfying the condition therein, we solve (78)
as
IA= 2 eλ
2
X
j=0
j+u1
X
k=0 λ
2k(j+m1)!
j!k!¯γ
¯γ+mj+m
.
(79)
2) Integral
IB
in (34): Following the same procedures as
in (78), we rewrite (34) as
IB=eλ
2
X
n=0
j+u1
X
q=0
X
i=0
i+m1
X
k=0 λ
2qai+kρi
i!k!n!q!ea γTγk
T
×Z
0
γm+n+i1exp γa ρ +m
¯γ+ 1dγ.
(80)
Similarly as we did in (79), we solve (80) as
IB=eλ
2
X
n=0
j+u1
X
q=0
X
i=0
i+m1
X
k=0 λ
2qai+kρi
i!k!n!q!
×(m+n+i1)!
a ρ +m
¯γ+ 1m+n+iea γTγk
T.
(81)
3) Integral ICin (35): Using (53), we rewrite (35) as
IC=eλ
2
X
n=0
n+u1
X
q=0
1
n!q!λ
2k
×ZγT
0
γm+n1exp γm
¯γ+ 1dγ.
(82)
Using [[18], Eq. (3.351/1)], where
Zz
0
xneµ xdx=n!
µn+1 eµz
n
X
k=0
n!
k!
zk
µnk+1
=µn1γ(n+ 1, µz),
[z > 0,Re µ > 0, n = 0,1,2,···],
(83)
we derive (82) as
IC=eλ
2
X
n=0
n+u1
X
q=0
1
n!q!λ
2q¯γ
¯γ+mm+n
×γm+n, γT¯γ+m
¯γ.
(84)
Substituting (79), (80) and (84) into (32), this concludes the
derivation.
APPENDIX G
ALTER NATI VE EXPRESSION FOR DUA L SSC
In this section, we will derive the expression in (38) by
evaluating
PD,S SC,2
in (32) for alternative expression as
follows.
1) Integral
IA
in (33): Let
x=2γ
, we rewrite (33) as
IA=4
2mZ
0
Qux, λxm1exp m x2
γdx. (85)
Using [[31], Eq. (8)], we solve (85) as
IA= 4
X
j=0
Γ(m+ju+j, λ
2
Γ(u+j)1 + m
¯γm+jj!
.(86)
2) Integral
IB
in (34): Using (52) and (53) for
Qu2γ, λ
and
Qm2a ρ γ, 2a γT
, we rewrite (34)
as
IB=
X
n=0
X
i=0
i+m1
X
k=0
Γu+n, λ
2
Γ(u+n)n!
ai+kρi
i!k!γk
Tea γT
×Z
0
γm+n+i1eγ(m
¯γ+a ρ+1)dγ.
(87)
Using (56) , we solve (87) as
IB=
X
n=0
X
i=0
i+m1
X
k=0
Γu+n, λ
2Γ(m+n+i)
Γ (u+n)m
γ++ 1m+n+in!i!k!
×ai+kρiγk
TeT.
(88)
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 13
3) Integral ICin (35): Using (52), we rewrite (35) as
IC=
X
p=0
Γu+p, λ
2
Γ(u+p)p!ZγT
0
γm+p1eγ(¯γ+m
¯γ)dγ. (89)
Using (83), we solve (89) for minteger values as
IC=
X
p=0
Γu+p, λ
2
Γ(u+p)p!¯γ+m
¯γ(m+p)
×γm+p, γT¯γ+m
¯γ.
(90)
Substituting (86), (88) and (90) into (32), this concludes the
derivation.
APPENDIX H
EXPRESSION FOR OPTIMAL THR ES HO LD
In this section, we will derive the expression in (39).
Employing Leibniz’s rule [[19], Eq. (3.3.7)] with the aid of
following identity given in [[29], Eq. (9)] as
∂β Qu(α, β ) = ββ
αu1
exp α2+β2
2Iu1(α β),
(91)
we rewrite (32) as
∂γ
T
PD,S SC,2=1
Γ(m)m
¯γm"ρ1m
22a γ
T
1m
2e
T
×Z
0
Qup2γ, λγm1
2eIm12apργ
Tγdγ
| {z }
I
Qup2γ
T,λγ
T
m1exp m γ
T
¯γ#.
(92)
To solve the integral
I
in (92), we perform changing variable
along with the aid of the series expansion of the modified
Bessel function given in [[18], Eq. (8.445)] as
Iν(z) =
X
k=0
1
Γ(ν+k+ 1) k!z
2ν+2 k.(93)
Then, we drive (92) as
∂γ
T
PD,S SC,2=1
Γ(m)m
¯γm"ρ1m
2p2a γ
Te
T
×
X
k=0
1
Γ(m+k)2m+kk!(aρ)m+2k1γ
T
k
×Z
0
Qux, λx2(m+k)1ea
2x2dx
| {z }
I
Qup2γ
T,λγ
T
m1exp m γ
T
¯γ#.
(94)
Using [[29], Eq. (29)] by following same procedures as in (50),
we can solve the integral Iin (94) as
IA=G0
1+1
2
u1
X
n=1 λ
2nΓ (m+k)
a+1
2m+kn!
×1F1m+k;n+ 1; λ
2 (a+ 1),
(95)
where
G0
1
can be obtained by evaluating the following integral
containing the first order of Marcum
Q
-function
Q(., .)
for
integer mvalues as
G0
1=Z
0
Qx, λx2(m+k)1ea
2x2dx. (96)
Using [[29], Eq. (25)], we evaluate
G0
1
for integer
m
values as
G0
1=2m+k1(m+k1)!
a2(m+k)1
a+ 1eλ
2
a
a+1
×"(1 + a)a
1 + am+k1
Lm+k1λ
2 (1 + a)
+
m+k2
X
n=0 a
a+ 1n
Lnλ
2 (a+ 1)#,
(97)
where Ln(.)denotes Laguerre polynomial of n-degree [18].
Substituting (95) into (94), this concludes the derivation.
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